Decision Modeling and Policy Management

Giampiero E.G. Beroggi

Preface

Decision modeling in policy management is a broad field that can be
addressed from different perspectives. Each perspective provides its own set
of tools, and the collection of these tools forms the arsenal of the policy
analyst. Our objective, however, is not to discuss as many methods and
techniques as possible, but, rather, to introduce a modeling paradigm for
addressing different aspects of decision analysis. The modeling paradigm
consists of a three-step decomposition of the analytic modeling process (see
Figure 1).

Figure 1: Three-step modeling process.

The first step is to translate a human's mental model of a real-world
decision problem into a structural model which illustrates diagrammatically
the elements of the decision problem and their relations. The elements of
decision making - actors, criteria, goals, uncertainty, and actions - are
depicted in form of an influence diagram, showing the relevant elements and
their mutual influence. In the second step, this problem structure is
conceptualized as a formal model. The formal model specifies the relations
between the elements and assigns values to them. The evaluations of the
decision options are represented in an evaluation table. In the third step,
based on the structural and the formal models, a resolution model is
defined. The resolution model describes how to solve the problem that has
been laid out in the first two steps.
Why this approach? Those familiar with visual modeling techniques know that
these three modeling steps are used over and over. Visual-interactive
software systems have emerged to support the analysis of systems, data, and
decision problems. These software packages are based on concepts such as
systems thinking, causal mapping, and visual modeling.

The subsequent question is: why not to discuss some of those modeling
paradigms which are embedded into commercial software packages? The reason
is that modeling approaches at the structural and formal levels are based on
specific resolution models. This bottom-up concept (from resolution model,
to formal model, to structural model) is often counter-intuitive, especially
when the structural level is used as a mean for communication with less
analytically skilled persons. For example, decision analysis packages do not
allow cycles or disconnected decision nodes in the structural model -
restrictions which are not conceptual, but, rather, are solely motivated by
the underlying formal and resolution models.

The added value of this text is the independent treatment of the three
modeling levels as part of the generic three-step modeling approach. The
elements of decision modeling are used to construct the structural model.
Each class of decision elements has its own icon. A basic structural model
is shown in Figure 2. Four locations are evaluated with three criteria.
These evaluations are done by three decision makers for four different
scenarios. The content goals show that safety and costs must meet certain
constraints, and the aspiration is to maximize benefits. The structural goal
says to choose two out of the four actions (locations) and to satisfy either
the safety or the cost constraint as defined (formalized) in the content
goals.

Figure 2: A basic structural model.

The structural model of Figure 2 indicates that the four alternatives must
be evaluated a total of 36 times (for three criteria, by three decision
makers, and for four scenarios). These evaluations are represented in an
evaluation matrix, in this case a four-dimensional matrix. The evaluation
matrix, however, represents objective measurements, expressed in terms of
safety, costs, and benefits. In this text, however, we will not address how
to measure, compute, or generate values for the evaluation matrix. Our
starting point is the evaluation matrix, and we will focus on how to
transform the evaluation matrix into subjective preference values, which
reflect the points of view of different decision makers.

Chapters I and II contain an introduction to the elements of decision
modeling and the three-step modeling approach. In the remaining eight
chapters, we address several principles of:

(1) transforming the objective evaluation matrix into a subjective
preference matrix,

(2) aggregating preferences over decision makers, criteria, and scenarios; and

(3) finding a solution, especially when the feasible alternatives are
defined implicitly.

The transformation of objective evaluation values into subjective preference
values has two major schools of thought: the descriptive and the normative
schools. In Chapters III and IV we discuss several descriptive preference
elicitation and aggregation methods, where the aggregation is done across
criteria.

In Chapter V we introduce a normative preference elicitation and
aggregation concept for multiple criteria. Before we extend this concept
for aggregation across scenarios in Chapter VII, we first introduce the
concepts of uncertainty in Chapter VI. Chapter VIII is dedicated to the
manipulation of probabilistic influence diagrams, using the concepts
introduced in the previous chapters.

In Chapter IX we address how to aggregate the assessments of multiple
decision makers and how to solve conflicts in case where no group-aggregated
assessment can be reached. Chapter X, finally, discusses how to aggregate
values within criteria and across alternatives and how to solve problems
with implicitly represented alternatives.

The discussions in this text will encompass several traditional topics in
decision analysis, including multicriteria methods, linear, non-linear, and
integer programming, value and utility theory, dynamic decision problems,
group decision problems, game theory, conflict resolution, etc. However,
because we use as our starting point this three-step modeling paradigm, we
will not lay out our discussion in terms of these techniques but in terms of
this three-step modeling paradigm. As a result, a consistent treatment of
several diverse topics is possible. Moreover, notations and definitions can
be kept to a minimum, and the different topics are addressed in the
appropriate context. For a more in-depth study of the specific analytic
methods and tools, appropriate references to classic works are made
throughout the text.

TABLE OF CONTENTS

Specification for Class 1996/97 viii

Preface xi

CHAPTER I: THE PROBLEM SOLVING PROCESS

1. The Context of Problem Solving 1

1.1 What is a Problem? 1
1.2 The Problem Solving Process 3

2. Problem Analysis: The Elements of Decision Making 1

2.1 Actors and Decision Makers 7
2.2 Criteria and their Decomposition and Aggregation 8
2.3 Goals and Objectives 9
2.4 Actions and Decision Variables 13

A) Explicit vs. Implicit Alternatives 14
B) Static vs. Dynamic Alternatives 15
C) Single vs. Multicriteria Alternatives 15
D) Pure vs. Mixed Action Alternatives 16

2.5 Uncertainties 16

3. Problem Definition 18

3.1 Evaluation Measures 18
3.2 Measurement Scales 20
3.3 Preference Elicitation 23
3.4 Binary Preference Relations 25

4. Problem Solution 27

4.1 Preference Aggregation 27
4.2 Dominant and Efficient Alternatives 28
4.3 Preference Graphs 29
4.4 Search for Solutions 32
4.5 Sensitivity Analysis 33

5. Summary 34

6. Questions 35

CHAPTER II: THE ANALYTIC MODELING PROCESS

1. From Problem to Model 37

1.1 The Model as Abstraction of Reality 37
1.2 The Analytic Modeling Process 38

2. Structural Models 40

2.1 Definitions and Elements 40
2.2 Actions 43
2.3 Criteria 44
2.4 Uncertainties 46
2.5 Actors 48

3. Formal Models 48

3.1 General Aspects 48
3.2 Descriptive vs. Normative Preference Elicitation 50
3.3 Single vs. Multiaction Decision Problems 53

4. Resolution Models 55

4.1 General Resolution Approach 55
4.2 Resolution Complexity 56

5. Interactive Complete Strong Preference Ordering 57

5.1 Structural Model 58
5.2 Formal Model 59
5.3 Resolution Model 59

6. Summary 62

7. Questions 63

CHAPTER III: DESCRIPTIVE ASSESSMENT - CRITERIA AND WEIGHTS

1. Relative Intensities and Weights 65

1.1 Consistent Assessment 65
1.2 Relative Importance and Ratio Scale 68
1.3 Resolution of Inconsistencies 70

2. Hierarchical Decomposition of Criteria 75

2.1 Structural Model 75
2.2 Formal Model 77

3. Aggregation of Criteria 78

3.1 Correlation and Multicollinearity 78
3.2 Computational Aspects 81
3.3 Principal Component Analysis 83
3.4 An Example of Dimension Reduction and Interpretation 86

4. Summary and Further Readings 89

5. Questions 90

CHAPTER IV: DESCRIPTIVE ASSESSMENT - ALTERNATIVES AND RANKING

1. Structural Models of Descriptive Approaches 92

1.1 Basic Concepts 92
1.2 Preference Aggregation: Basic Principles 94

2. Formal Models for Descriptive Approaches 96

2.1 Relative and Absolute Preferences 96

A) Relative Preference Assessments 97
B) Absolute Preference Assessments 98

2.2 Ordinal Scale in Descriptive Dominance Assessment 100
2.3 Interval and Ratio Scales in Descriptive Dominance Assessment 101
2.4 Formalization of Preference Aggregation 102
2.5 Preference Score Cards 105

2.6 Evaluating Infrastructure Projects 107
2.7 Formalizing Structural and Content Goals 110

3. Resolution Models 112

3.1 Complete Preference Ranking of Alternatives 112
3.2 Incomplete Preference Ranking of Alternatives 113
3.3 Sensitivity Analysis and Rank Preservation 117

A) Structural Instability 118
B) Functional Instability 120
C) Numerical Instability 122

4. Summary and Further Readings 123

5. Questions 124

CHAPTER V: VALUES AND NORMATIVE CHOICE

1. The Structural Model 125

1.1 Conceptual Aspects 125
1.2 The Structure 128

2. The Formal Model 129

2.2 Motivation and Axioms of Value Theory 129
2.2 Preferential Independence 132

A) Two Criteria 133
B) Three Criteria 134
C) More than three Criteria 135

2.3 Additive Multicriteria Value Functions 136
2.4 Interpretation of 2D Value Functions 140

3. The Resolution Model 142

3.1 The General Approach 142
3.2 Assessment of Mutual Preferential Independence 143
3.3 Elicitation of Conditional Value Functions 146
3.4 Assessment of Scaling Constants 149
3.5 Evaluation of Alternatives 150

A) Convex Set and Efficient Frontier 150
B) Discrete Decision Options 152
C) Continuous Decision Options 152

3.6 Sensitivity Analysis 154
3.7 Comments about Multicriteria Subjective Value Functions 154

4. Summary and Further Readings 156

5. Questions 157

CHAPTER VI: CHOICES UNDER UNCERTAINTY

1. Decision Making Under Complete Uncertainty 159

1.1 Structural Model 159
1.2 Formal Model 160

A) Wald's MaxMin Rule 161
B) Savage's MinMax Regret Rule 162
C) Hurwicz's Optimism-Pessimism Index 163
D) Laplace's Principle of Insufficient Reasoning 163

1.3 Resolution Model 164

2. Decision Making Under Risk 165

2.1 Structural Model 165
2.2 Formal Model 166
2.3 Concepts of Probability Theory 166
2.4 Decision Rules 177

A) Individual Risk 177
B) Collective Risk 178
C) Group Risk 180

2.5 Aggregation of Linguistic Variables 182
2.6 Resolution Model 186

3. Summary and Further Readings 187

4. Questions 188

CHAPTER VII: UNCERTAINTY AND NORMATIVE CHOICE

1. The Structural Model for Decision Making Under Uncertainty 190

1.1 Conceptual Aspects 190
1.2 The Structural Model 193

2. The Formal Model of Utility Theory 194

2.1 Motivation of Utility Theory 194
2.2 The Axioms of Utility Theory 195
2.3 Assessing Conditional Utility Functions 198
2.4 Interpretation of the Utility Function and Risk Attitudes 203
2.5 Utility Independence of Criteria 204
2.6 Multicriteria Utility Functions 206

A) Multilinear Multicriteria Utility Functions 206
B) Multiplicative Multicriteria Utility Functions 207
C) Additive Multicriteria Utility Functions 208

3. The Resolution Model 209

3.1 The General Approach 209
3.2 Assessment of Mutual Utility Independence 212
3.3 Assessment of Scaling Constants and Selection of Model 213
3.4 Assessment of Conditional Utility Functions and
      Evaluation of Alternatives 215
3.5 Comments about Multicriteria Utility Functions 215

4. Summary and Further Readings 217
5. Questions 218

CHAPTER VIII: SEQUENTIAL DECISION MAKING

1. The Structure of Sequential Decisions 221

1.1 Concept of Probabilistic Influence Diagrams 221
1.2 The Meaning of Influences in Probabilistic Influence Diagrams 223

2. The Formal Model 229

2.1 Defining Probabilistic Relations 229
2.2 Decision Trees 230
2.3 Probabilistic Influence Diagrams 231

3. The Resolution Model 234

3.1 Resolution Steps 234

A) Barren Node Removal 236
B) Arc Reversal 236
C) Chance Node Removal 239
D) Decision Node Removal 241

3.2 Node Elimination Algorithm 243

4. Sensitivity Analysis 246

4.1 Probabilistic Sensitivity 246
4.2 Value of Information 247

5. Summary and Further Readings 248

6. Questions 249

CHAPTER IX: MULTI-ACTOR DECISION MAKING

1. Structural Models in Multi-Actor Settings 253

1.1 Conceptual Aspects 253
1.2 Finding the Best Alternative under Group Decision Making 253
1.3 Find the Best Alternative for each Individual Actor 254

2. Group Decision Making 255

2.1 Aggregation of Ordinal Assessments 255

A) Distance Measures 255
B) Statistical Approaches 257

2.2 Aggregation of Cardinal Assessments 261

A) Aggregation Process 261
B) Aggregation of Normative Evaluations 262
C) Aggregation of Descriptive Evaluations 264

2.3 Aggregation of Linguistic Assessments 267

A) Aggregation across Criteria 268
B) Aggregation across Decision Makers 269

3. Conflict Resolution 270

3.1 Two-Actor Strictly Competitive Settings 270

A) Pure Strategy Sequential Elimination 271
B) Pure Strategy MinMax Test 272
C) Mixed (Randomized) Strategy Solutions 273

3.2 Conflict Resolution and Negotiation Support 275

A) Non-Pareto Optimal Equilibrium 276
B) Pure and Mixed Strategy Equilibrium 277

B.1) MaxMin Approach 278
B.2) Estimated Uncertainty 278

C) Cooperation in Conflict Situations 279
D) Bargaining (Negotiation) Sets 280

4. Summary and Further Readings 282
5. Questions 283

CHAPTER X: CONSTRAINT-BASED POLICY OPTIMIZATION

1. Structural Model 285

1.1 Basic Concepts and Types of Decision Problems 285
1.2 The Structural Model 286

2. Formal Model 287

2.1 Content Goals: Constraints and Aspiration Levels 287

A) Single-Criterion Optimization 287
B) Multi-Criteria Optimization 293

2.2 Structural Goals: Actions and Content Goals 295

A) Structural Goals referring to Actions 295
B) Structural Goals referring to Content Goals 297

2.3 Additional Examples of Formal Models 299

A) Transportation Planning 299
B) Two-Actor Zero-Sum Conflicts 300
C) Nurse Scheduling 302
D) Mixing Substances 304

3. Resolution Models 305

3.1 Real-Valued Decision Variables 305

A) Theoretical Considerations 305
B) The Simplex Algorithm 306
C) Numerical Example 312
D) Comments about the Simplex Algorithm 314

3.2 Duality and Sensitivity Analysis 315

A) General Concept 315
B) Numerical Example 316

3.3 Integer Decision Variables 318
3.4 Binary Decision Variables 319
3.5 Computer Implementation 321

4. Summary and Further Readings 323
5. Questions 324

References 326

Symbol Index 333

Subject Index 335

 RETURN TO DECISION ANALYSIS SYLLABI